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probability - LECTURE NOTES MEASURE THEORY and PROBABILITY...

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LECTURE NOTES MEASURE THEORY and PROBABILITY Rodrigo Ba˜nuelos Department of Mathematics Purdue University West Lafayette, IN 47907 June 20, 2003
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2 I SIGMA ALGEBRAS AND MEASURES § 1 σ –Algebras: Definitions and Notation. We use Ω to denote an abstract space. That is, a collection of objects called points. These points are denoted by ω . We use the standard notation: For A, B Ω, we denote A B their union, A B their intersection, A c the complement of A , A \ B = A - B = { x A : x B } = A B c and A Δ B = ( A \ B ) ( B \ A ). If A 1 A 2 , . . . and A = n =1 A n , we will write A n A . If A 1 A 2 . . . and A = n =1 A n , we will write A n A . Recall that ( n A n ) c = n A c n and ( n A n ) c = n A c n . With this notation we see that A n A A c n A c and A n A A c n A c . If A 1 , . . . , A n Ω, we can write n j =1 A j = A 1 ( A c 1 A 2 ) ( A c 1 A c 2 A 3 ) . . . ( A c 1 . . . A c n - 1 A n ) , (1.1) which is a disjoint union of sets. In fact, this can be done for infinitely many sets: n =1 A n = n =1 ( A c 1 . . . A c n - 1 A n ) . (1.2) If A n , then n j =1 A j = A 1 ( A 2 \ A 1 ) ( A 3 \ A 2 ) . . . ( A n \ A n - 1 ) . (1.3) Two sets which play an important role in studying convergence questions are: lim A n ) = lim sup n A n = n =1 k = n A k (1.4) and lim A n = lim inf n A n = n =1 k = n A k . (1.5)
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3 Notice ( lim A n ) c = n =1 k = n A n c = n =1 k = n A k c = n =1 k = n A c k = lim A c n Also, x lim A n if and only if x k = n A k for all n . Equivalently, for all n there is at least one k > n such that x A k 0 . That is, x A n for infinitely many n . For this reason when x lim A n we say that x belongs to infinitely many of the A n s and write this as x A n i.o. If x lim A n this means that x k = n A k for some n or equivalently, x A k for all k > n . For this reason when x lim A n we say that x A n , eventually . We will see connections to lim x k , lim x k , where { x k } is a sequence of points later. Definition 1.1. Let F be a collection of subsets of Ω. F is called a field (algebra) if Ω ∈ F and F is closed under complementation and finite union. That is, (i) Ω ∈ F (ii) A ∈ F ⇒ A c ∈ F (ii) A 1 , A 2 , . . . A n ∈ F ⇒ n j =1 A j ∈ F . If in addition, (iii) can be replaced by countable unions, that is if (iv) A 1 , . . . A n , . . . ∈ F ⇒ j =1 A j ∈ F , then F is called a σ –algebra or often also a σ –field . Here are three simple examples of σ –algebras. (i) F = {∅ , Ω } ,
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4 (ii) F = { all subsets of Ω } , (iii) If A Ω , F = {∅ , Ω , A, A c } . An example of an algebra which is not a σ –algebra is given by the following. Let Ω = R , the real numbers and take F to be the collection of all finite disjoint unions of intervals of the form ( a, b ] = { x : a < x b } , -∞ ≤ a < b < . By convention we also count ( a, ) as right–semiclosed. F is an algebra but not a σ –algebra. Set A n = (0 , 1 - 1 n ] .
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